Here are the positions at three different times for a bee in flight (a bee’s top speed is about 7 m/s ). $$\begin{gathered} \mathbf{Time}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{6}\mathbf{.}\mathbf{3}\mathbf{}\mathbf{s}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{6}\mathbf{.}\mathbf{8}\mathbf{}\mathbf{s}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{7}\mathbf{.}\mathbf{3}\mathbf{}\mathbf{s} \\ \mathbf{}\mathbf{Position}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{<}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{,}\mathbf{9}\mathbf{.}\mathbf{4}\mathbf{,}\mathbf{0}\mathbf{>}\mathbf{m}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{3}\mathbf{,}\mathbf{6}\mathbf{.}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{>}\mathbf{m}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{<}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{,}\mathbf{1}\mathbf{.}\mathbf{7}\mathbf{,}\mathbf{0}\mathbf{>}\mathbf{m} \end{gathered}$$(a)Between 6.3 s and 6.8 s , what was the bee’s average velocity? Be careful with signs.

(b)Between6.3 s and 7.3 s, what was the bee’s average velocity? Be careful with signs.

(c)Of the two average velocities you calculated, which is the best estimate of the bee’s instantaneous velocity at time 6.3 s ?

(d)Using the best information available, what was the displacement of the bee during the time interval from 6.3 s to 6.33 s ?

Position of the bee at $t_1=6.3\mathrm{s}:<-3.5,9.4,0>\mathrm{m}$

Position of the bee at $t_2=6.8\mathrm{s}:<-1.3,6.2,0>\mathrm{m}$

Position of the bee at$t_3=7.3\mathrm{s}:<0.5,1.7,0>\mathrm{m}$

The position vector of a location $<x_1,y_1,z_1>$with respect to another location $<x_2,y_2,z_2>$is

$\overrightarrow{\mathbf{r}}\mathbf{=}\left(x_2-x_1\right)\hat{\mathbf{i}}\mathbf{+}\left(y_2-y_1\right)\hat{\mathbf{j}}\mathbf{+}\left(z_2-z_1\right)\hat{\mathbf{k}}.....\left(\mathrm{l}\right)$

The average velocity for a displacement $\overrightarrow{r}$in time $t$ is

$\overrightarrow{\mathbf{v}}\mathbf{=}\frac{\overrightarrow{\mathbf{r}}}{\mathbf{t}}\mathbf{}\mathbf{}\mathbf{}.....\left(\mathrm{ll}\right)$

The displacement of the bee from time $t_1$to time $t_2$ is the position vector of the location of the bee at time $t_2$with respect to the location of the bee at time $t_1$. From equation (I), this displacement is

$$\begin{gathered} {\overrightarrow{r}}_{21}=\left\{\left(-1.3-\left(-3.5\right)\right)\hat{i}+\left(6.2-9.4\right)\hat{j}+\left(0-0\right)\hat{k}\right\}\mathrm{m} \\ =\left(2.2\hat{\mathrm{i}}-3.2\hat{\mathrm{j}}\right)\mathrm{m} \end{gathered}$$

This displacement took place in time $t_2-t_1$. From equation (II), the average velocity is

$$\begin{gathered} {\overrightarrow{v}}_{21}=\frac{{\overrightarrow{r}}_{21}}{t_2-t_1} \\ =\frac{\left(2.2\hat{i}-3.2\hat{j}\right)\mathrm{m}}{\left(6.8\mathrm{s}-6.3\mathrm{s}\right)} \\ =\frac{\left(2.2\hat{i}-3.2\hat{j}\right)\mathrm{m}}{0.5\mathrm{s}} \\ =\left(4.4\hat{i}-6.4\hat{j}\right)\mathrm{m}/\mathrm{s} \end{gathered}$$

Thus, the required average velocity is $\left(4.4\hat{i}-6.4\hat{j}\right)\mathrm{m}/\mathrm{s}$.

The displacement of the bee from time $t_1$to time $t_3$ is the position vector of the location of the bee at time $t_3$ with respect to the location of the bee at time $t_1$. From equation (I), this displacement is

$$\begin{gathered} {\overrightarrow{r}}_{31}=\left\{\left(0.5-\left(-3.5\right)\right)\hat{i}+\left(1.7-9.4\right)\hat{j}+\left(0-0\right)\hat{k}\right\}\mathrm{m} \\ =\left(4\hat{\mathrm{i}}-7.7\hat{\mathrm{j}}\right)\mathrm{m} \end{gathered}$$

This displacement took place in time $t_3-t_1$. From equation (II), the average velocity is

$$\begin{gathered} {\overrightarrow{v}}_{31}=\frac{{\overrightarrow{r}}_{31}}{t_3-t_1} \\ =\frac{\left(4\hat{i}-7.7\hat{j}\right)\mathrm{m}}{\left(7.3\mathrm{s}-6.3\mathrm{s}\right)} \\ =\frac{\left(4\hat{i}-7.7\hat{j}\right)\mathrm{m}}{1\mathrm{s}} \\ =\left(4\hat{i}-7.7\hat{j}\right)\mathrm{m}/\mathrm{s} \end{gathered}$$

Thus, the required average velocity is $\left(4\hat{i}-7.7\hat{j}\right)\mathrm{m}/\mathrm{s}$.

More the bee travels, more will its velocity change. So the average estimate calculated at a small time interval will be a better estimate of instantaneous velocity. Hence ${\overrightarrow{v}}_{21}$ will be a better estimate of the bee's instantaneous velocity at 6.3 s .

Consider the velocity of the bee to remain constant at ${\overrightarrow{v}}_{21}$in between 6.3 s and $t_4=6.33\mathrm{s}$. From equation (II), the displacement in this time period is

Thus, the required displacement is $\left(0.132\hat{\mathrm{i}}-0.192\hat{\mathrm{j}}\right)\mathrm{m}$.